930 science: The Middle East added Some of these tables also provide the square of the principal number The Babylonians had no division tables, but they did make extensive use of reciprocal tables (A reciprocal is simply divided by the number in question, so that the reciprocal of 60 is 1/60.) Thus, instead of dividing, they multiplied by a reciprocal To aid them in this task, they created extensive reciprocal tables There are hundreds of these tables, along with tables for squares, square roots, and cubes, many of them compiled by students Finally, there are coefficient lists, which list conversion factors used in geometry (for example, the ratio of a diagonal to a square’s side) and weights and measures problems In addition to these tables are so-called problem texts, that is, exercises used in schools Some of the tablets that record these texts contain a number of problems on a single topic; others contain problems related to different topics They have given historians insight into the kinds of problems examined in schools Nearly all of the problems ask students to come up with a number, never with any kind of proof Thus, they are “algebraic” problems, in contrast to the “geometric” problems that ancient Greek students studied (In algebra the goal is to arrive at a correct answer, expressed as a number; in geometry the goal is often to prove, for example, that the angles of a triangle add up to 360 degrees) Interestingly, modern mathematicians have taken a more extensive interest in Babylonian mathematics because of its emphasis on algorithms, or following a process to arrive at an answer Most such problems were what modern students know as “story problems,” that is, problems couched in everyday terms by calling for calculations of things in the real world: the length of a canal or broken reeds, the weight of a quantity of stones, or the number of bricks used in a building Many of the problems are complex, requiring students to use not just single equations but more complex linear and quadratic equations Some of the texts record the procedures students would follow to solve the problem, but none state general principles Instead, the emphasis seems to have been on working problems as examples often enough so that the student could then work a new problem with different values In many cases, all the problems in a group have the same answer, suggesting that the group of exercises was designed to teach a process of computation rather than to arrive at a correct answer Although the Babylonians did not have an organized system of geometry—they did not compute angles, for instance—many of the problems have geometric implications, and many contain drawings of squares, rectangles, circles, and triangles Some problems required the student to compute such quantities as lengths of diagonals or sides or to calculate volume or area Bricks and calculating a quantity of bricks seem to have been a preoccupation with Babylonian math teachers For reasons that historians not fully understand, the mathematical record of the Old Babylonians comes to an abrupt halt after about 1600 b.c.e What follows is a thou- sand-year gap in the record After this millennium the record resumes, and historians have records of continued interest in mathematics in the Mesopotamian region ASTRONOMY During the first millennium b.c.e the Babylonians compiled columnar lists of stars, with the columns listing the stars and their positions in relation to the positions of other stars So great became the interest in observation of the stars that temples became astronomical observatories It should be noted that at the same time, the Assyrians of the Near East also gave great importance to astronomy, and Assyria’s capital, Kalhu (modern Nimrud), was a center of astronomical observation and study In both Babylon and Assyria, lists of such events as eclipses were meticulously kept after about 800 b.c.e.—so accurately that future eclipses could be and were predicted Again, the Babylonians lacked a system, an underlying theory to explain and systematize the phenomena they observed in the heavens Only after about the sixth century b.c.e did they begin to organize their knowledge into a system They did so only because they began working alongside Greek astronomers, who seemed, in common with the ancient Greeks, generally to have been better equipped to think systematically Many of the famous Babylonian astronomers from this period even took Greek names Naburimanni, who lived around 500 b.c.e., became called Naburianos, and the later astronomers Kidinnu and Belussur became, respectively, Cidenas and Berossus Babylonian astronomical observation was not, however, a “scientific” endeavor in the sense of an activity intended to develop an understanding of the atmosphere or the planets and the stars Rather, the purpose of astronomical observation was completely different than it is today It was employed in divination, akin to modern and medieval astrology Celestial divination was used to foretell the fate not of individuals but of kings and the state The decision to launch a military campaign, like almost all major decisions, was almost always preceded by celestial or some other form of divination MEDICINE Much of the interest in the natural sciences among the ancient Mesopotamians focused on medicine and healing, an early effort to exert some human control over inexplicable forces In the ancient Akkadian language, the word for doctor is literally translated as “fluids expert.” Over millennia doctors in the region accumulated a vast store of information having to with drugs, salves, and other medicines either taken orally or applied to the body Most of these drugs consisted either of minerals or plant extracts, including spices The practice of medicine was closely interwoven with magic and the science of omens The application of drugs was typically accompanied by prayers and incantations The goal was to strengthen the patient’s will to recover Interestingly, modern medicine has come to recognize that a patient’s frame of mind can play an important role in recovery; 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